Define variables

How deep is your love?

Define a variable for each construct of the VAST display. These can be measured variables or unmeasured (mediating) variables.

You can either refer to an actual measurement procedure or simply define a variable. In both cases you should explicitly define the following properties:

• variable name (i.e., to which concept in your VAST display does it refer?),
• scale level (categorical, ordinal, interval scale),
• range of possible values (e.g., 0 … 1),
• semantic anchors (e.g., 0 = “complete absence”, 1 = “maximal possible value”)

Define variables

Some guiding questions and heuristics

• Type of variable: continuous, dichotomous, ordinal, categorical?
• Scale level of measurement (Stevens’s typology):
  Nominal → Ordinal → Interval → Ratio
• Is the variable naturally bounded? On both sides or only one?
• How can the numbers be interpreted?
  • Natural/objective scale (e.g. physical distance)
  • As standardized z-scores?
  • Normalized to a value between 0 and 1? Or rather -1 to 1?
  • Can we find an empirical or semantic calibration?
    • Just noticable difference
    • 100 = “largest realistically imaginable quantity of that variable”

Group work (15 min.): Specify variables

Group work (10 min.)

Sketch a first functional relationship on paper

We want to model the following phenomenon (a specific version of the bystander effect):

1. Without other people present, the tendency (probability or frequency) that a person helps a victim is high.
2. The tendency of helping a victim decreases monotonically with the number of other people (bystanders) present.
3. The tendency of helping a victim never drops to 0.

Task: Sketch a first functional relationship that could model this phenomenon. Use the variables you defined in the previous step (including their labels and ranges).

Step 2: Define functional relationships between variables

Every causal path needs to be implemented as a mathematical function, where the dependent variable/output \(y\) is a function of the input variable(s) \(x_i\).

\(y = f(x_1, x_2, ..., x_i)\)

This can, for example, be a linear function, \(y = \beta_0 + \beta_1x_1\).

Step 2: Define functional relationships between variables

Fixed and free parameters

Step 2: Define functional relationships between variables

Fixed and free parameters

Fixing a parameter:

\(\color{forestgreen} \beta_0 \color{black} = 1 \rightarrow \color{red} y = \color{forestgreen} 1 \color{black} + \color{forestgreen} \beta_1 \color{blue} x\)

That means, the slope \(\color{forestgreen} \beta_1\) still can vary, but the intercept is fixed to 1.

Discussion

Sketch a first functional relationship on paper

• Which aspects of your sketched function could be free parameters? Describe them in plain language.
• Draw a couple of alternative plots that (a) follow the same functional form and (b) also fullfill the criteria of the phenomenon.
• What is the semantic meaning of the free parameters?

Tool 1: The logistic function family

As a linear function is unbounded, it can easily happen that the computed \(y\) exceeds the possible range of values.

If \(y\) has defined boundaries (e.g., \([0; 1]\)), a logistic function can bound the values between a lower and an upper limit (in the basic logistic function, between 0 and 1):

\(y = \frac{1}{1 + e^{-x}}\)

Tool 1: The logistic function family

logistic <- function(x, d = 0, a = 1, min = 0, max = 1) {
 min + (max - min)/(1 + exp(a * (d - x)))
}

Tool 1: The logistic function family

(basic logistic function as dotted grey line)

Tool 1: The logistic function family

The d, a, min, and max parameters can be used to “squeeze” the S-shaped curve into the range of your variables. For example, if your \(x\) variable is defined on the range \([0; 1]\), the following function parameters lead to a reasonable shift:

Use the full mathematical toolbox

Of course, the logistic function and the beta distribution are just a start - you can use the full toolbox of mathematical functions to implement your model!

Excursus: Functions in R

• What is a function?
  • A reusable block of code designed to perform a specific task.
  • Can be an implementation of an actual mathematical function \(y = f(x, z)\)
  • But can also be a more complex operation, like reading a file, transforming data, or plotting a graph.
• Why use functions?
  • Modularity: Break down complex problems into manageable parts.
  • Reusability: Write once, use multiple times.
  • Maintainability: Easier to debug and update code.

Anatomy of an R Function

# Basic structure of a function
my_function <- function(arg1, arg2) {
  # Function body
  result <- arg1 + arg2
  return(result)
}

# use meaningful names

sum <- function(x, y) {
  S <- x+y
  return(S)
}

# short version: the last computation 
# is returned by default

sum <- function(x, y) {x+y}

Creating and Using Your Own Function

1. Define the Function:

get_y <- function(x) {
  y <- x^2
  return(y)
}

1. Call the Function:

y <- get_y(x=2)
print(y)  # Output: 4

Tips:

Example with roxygen2 documentation

#' Compute the updated expected anxiety
#'
#' The expected anxiety at any given moment is a weighted average of 
#' the momentary anxiety and the previous expected anxiety.
#'
#' @param momentary_anxiety The momentary anxiety, on a scale from 0 to 1
#' @param previous_expected_anxiety The previous expected anxiety, on a scale from 0 to 1
#' @param alpha A factor that shifts the weight between the momentary anxiety (alpha=1) 
#'              and the previous expected anxiety (alpha=0).
#' @return The updated expected anxiety, as a scalar on a scale from 0 to 1

get_expected_anxiety <- function(momentary_anxiety, previous_expected_anxiety, alpha=0.5) {
  momentary_anxiety*alpha + previous_expected_anxiety*(1-alpha)
}

Exercise: Document our exponential decay function

Check out roxygen2 and document our exponential decay function with:

• title
• description
• parameters
• return value.

get_p_help <- function(NOPB, y_0, y_final, BSE_strength) {
  p_help <- y_final + (y_0 - y_final) * exp(-BSE_strength * NOPB)
  return(p_help)
}

Apply everything to reduced model

Step 3: Implement the functions in R

Put everything together

Connect all functions to one “super-function”, which takes all exogenous variables as input and computes the focal output variable(s).

Test the super-function:

• Enter some reasonable combinations of parameters
• Draw plots where you vary one parameter on the x-axis and see the behavior of the output variable on the y-axis.

Step 4 (Excursus): Fitting the functions to empirical data

We can tune our free parameters to fit the model as good as possible to empirical data. This is called model fitting.

We use the empirical data in

Step 5: Simulate a full data set

Which input variables do your simulated participants bring to the experiment? Real participants bring variability in these variables.

Assume a specific distribution of all input variables that participants bring to the experiment. Even better: find empirical evidence for it, for example from existing open data sets or summary statistics from publications.

Maybe you need to transform the scale of the empirical variable into the scale of your simulated equivalent (e.g., via z-standardization).

Tool 2: The Distribution Zoo

If you start simulating data for your virtual participants, you want to simulate their starting values yb drawing random values from a distribution. For example, the virtual participants might differ in their anxiety, which you previously defined on the range \([0; 1]\).

How can you generate random values that roughly look like a normal distribution, but are bounded to the defined range?

For simulations, it is good to know some basic distributions. Here are three interactive resources for choosing your distribution:

• The Distribution Zoo by Ben Lambert and Fergus Cooper
• The Probability Distribution Explorer by Justin Bois
• Interactive collection of distributions by Richard Morey

Tool 3: The beta distribution

A handy distribution for the \([0; 1]\) range is the beta distribution. With its two parameters \(\alpha\) (also called \(a\) or shape1) and \(\beta\) (also called \(b\) or shape2), it can take many different forms:

Tool 3: The beta distribution

How to choose \(\alpha\) and \(\beta\)? Asking ChatGPT/Wolfram Alpha for assistance

Assume that you intuitively started with a normal distribution with \(M=0.2\) and \(SD=0.1\) (rnorm(mean=0.2, sd=0.1)) to simulate your values. But this function can generate values < 0 and > 1.

How can you achieve a beta distribution that approximates the properties of this normal distribution? You can work out the math yourself (e.g., by looking at the formula for the mean and the variance of the beta distribution) - or you can ask ChatGPT. After all, we only use this function as a tool to get some plausible values.

Tool 3: The beta distribution

Tool 3: The beta distribution

You can generate random values in R with the rbeta function. Here’s a comparison of a normal distribution and a matched beta distribution that respects the boundaries \([0; 1]\):

x.random <- rnorm(10000, mean=0.2, sd=0.1)
hist(x.random, xlab = "x", ylab="", main="Normal distribution (M=0.2, SD=0.1)", xlim=c(-0.3, 1.1))

x.beta <- rbeta(10000, shape1=3, shape2=12)
hist(x.beta, xlab = "x", ylab="", main="beta distribution (a=3, b=12)", xlim=c(-0.3, 1.1))

Tool 3: The beta distribution

Approximating a normal distribution with a beta distibution

Step 5: Simulate a full data set

Once all of these values are defined, simulate a sample of participants (with their starting values), and put half of them into the experimental group, and half of them into the control group. Use the resulting output values to computer a regular t-test.

Now the questions are:

• Does the simulated model produce the phenomenon? How large is the effect size?
• What happens if you change some of the functional parameters? Is the phenomenon still there? Is there a point in parameter space where the phenomenon breaks down?

Contact

• @nicebread@scicomm.xyz
• ed.uml.ysp@tdorbneohcs.xilef
• https://www.nicebread.de
• https://github.com/nicebread